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Discrete-Time Systems
Denis Efimov, Wilfrid Perruquetti, Tarek Raissi, Ali Zolghadri
To cite this version:
Denis Efimov, Wilfrid Perruquetti, Tarek Raissi, Ali Zolghadri. On Interval Observer Design for Time- Invariant Discrete-Time Systems. European Control Conference 2013, Jul 2013, Zurich, Switzerland.
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On Interval Observer Design for Time-Invariant Discrete-Time Systems
Denis Efimov, Wilfrid Perruquetti, Tarek Raïssi, Ali Zolghadri
Abstract—The problem of interval state observer design is addressed for time-invariant discrete-time systems. Two solutions are proposed: the first one is based on a similarity transformation synthesis, which connects a constant matrix with its nonnegative representation ensuring the observation error positivity. The second contribution shows that in discrete-time case the estima- tion error dynamics always can be represented in a cooperative form without a transformation of coordinates. The corresponding observer gain can be found as a solution of the formulated LMIs.
The performances of the proposed observers are demonstrated through computer simulations.
I. INTRODUCTION
The observer design problem is very challenging and its solution is demanded in many applications [1], [2], [3]. There exist many approaches dealing with the design techniques for state observers. In some cases, due to disturbance or uncertain parameter presence the synthesis of a conventional estimator (converging in the noise-free case to the ideal value of the state) is not possible. However, an interval estimation still may be feasible. By interval (or set-membership) estimation we understand an observer that, using input-output measurements, evaluates the set of admissible values (interval) for the state at each instant of time.
There exist many interval observers proposed for continuous-time (linear and nonlinear) systems based on the monotone system theory [4], [5], [6], [7]. One of the most complex assumptions for the interval observer design deals with cooperativity/monotonicity of the interval estimation error dynamics. Recently it was shown that under some mild conditions, applying similarity transformation, a Hurwitz matrix can be transformed to a Hurwitz and Metzler (cooperative) one [6], [8], [7]. In [6], [8] this transformation is time-varying, in [7] the transformation matrix is constant and real, it is a solution of the Sylvester equation (a constructive procedure for this solution calculation was also given in [7]). In [9] this result has been extended to the class of linear time-varying systems, when constant and time-varying similarity transformations have been proposed representing an interval matrix (a time-varying matrix) in the Metzler form.
The first two authors are with the Non-A project at INRIA - LNE, Parc Scientifique de la Haute Borne, 40 avenue Halley, Bât.A Park Plaza, 59650 Villeneuve d’Ascq, France,{denis.efimov;wilfrid.perruquetti}@inria.fr.
The third author is with Conservatoire National des Arts et Métiers (CNAM), Cedric - laetitia 292, Rue St-Martin, 75141 Paris Cedex 03, tarek.raissi@cnam.fr.
The last author is with University of Bordeaux, IMS-lab, Automatic control group, 351 cours de la libération, 33405 Talence, France,Ali.Zolghadri@ims- bordeaux.fr.
Several set-membership state estimators have been devel- oped for discrete-time models in the the literature using simple geometrical forms such as parallelotopes, ellipsoids, zonotopes or intervals [10], [11], [12], [13], [14], [15]. They are based on the well-known prediction/correction approach (also called open-loop observers, framers or predictors, then the system equations are solved starting from a set of initial conditions taking on each step the values consistent with the output measurements). The main drawback of this approach is that the convergence rate cannot be tuned since it is not based on an observer gain. As an alternative, we propose to extend the interval observer methodology, initially developed for continuous-time systems [4], [8], [7], to uncertain discrete- time systems.
In many cases the measurements are available at the discrete instants of time, then the discretized models of plants are used.
It is interesting to note that a cooperative continuous-time system remains cooperative in the discrete time under (Euler) discretization. The objective of this work is to develop the ap- proach of interval observer design to the discrete-time systems.
The case of Linear Time-Invariant (LTI) models is studied (it is assumed that nonlinear terms depend on measurable input- output signals and time). First, a similarity transformation of a Schur stable matrix to a Schur stable and nonnegative matrix is designed. Second, it is shown that the estimation dynamics can be always represented in a cooperative form without a transformation of coordinates, the observer gain in this case is a solution of LMIs. An academic planar system and the Hénon map system are considered in this work as examples.
The paper is organized as follows. Some basic facts from the theory of interval estimation are given in Section 2. The main result is described in Section 3. Example of computer simulation is presented in Section 4.
II. PRELIMINARIES
The real and integer numbers are denoted as R and Z respectively, R+ = {τ ∈ R : τ ≥ 0} and Z+ = Z∩R+. Euclidean norm of a vector x∈ Rn will be denoted as |x|, and for a measurable and locally essentially bounded input u:Z→Rthe symbol||u||[t0,t1] denotes itsL∞ norm:
||u||[t0,t1]= sup{|u(t)|, t∈[t0, t1]},
if t1 = +∞ then we will simply write||u||. We will denote as L∞ the set of all inputs u with the property ||u|| < ∞.
Denote the sequence of integers 1, ..., kas 1, k. The symbols In andEn×m denote the identity matrix and the matrix with all entries equal 1 respectively (with dimensions n×n and
R the vector of its eigenvalues is denoted as λ(A). The relation P 0 (P 0) means that the matrixP∈Rn×n is positive (nonnegative) definite.
A. Interval analysis
For two vectors x1, x2 ∈Rn or matricesA1, A2 ∈Rn×n, the relationsx1 ≤x2 and A1 ≤A2 are understood elemen- twise. Given a matrix A ∈Rm×n define A+ = max{0, A}, A−=A+−A.
Lemma 1. Letx∈Rn be a vector variable, x≤x≤xfor somex, x∈Rn, and A∈Rm×n be a constant matrix, then
A+x−A−x≤Ax≤A+x−A−x. (1) Proof:Note thatAx= (A+−A−)x, that forx≤x≤x gives the required estimates.
B. Cooperative discrete-time linear systems
A matrix A∈Rn×n is called Schur stable if all its eigen- values have the norm less than one, it is called nonnegative if all its elements are nonnegative. Any solution of the linear system
x(t+ 1) =Ax(t) +ω(t), ω:Z+→Rn+, t∈Z+, withx∈Rn and a nonnegative matrixA∈Rn×n+ , is elemen- twise nonnegative for all t≥0 provided thatx(0)≥0 [16].
Such dynamical systems are called cooperative (monotone) [16].
Lemma 2. [7] Given the matrices A ∈ Rn×n, R ∈ Rn×n and C ∈Rp×n.If there is a matrix L∈Rn×p such that the matricesA−LCandRhave the same eigenvalues, then there is aS∈Rn×n such thatR=S(A−LC)S−1 provided that the pairs(A−LC, e1)and (R, e2)are observable for some e1∈R1×n,e2∈R1×n.
This result was used in [7] to design interval observers for LTI systems with a Metzler matrix R (the matrix R is called Metzler if all its off-diagonal elements are nonnegative).
The main difficulty is to prove the existence of a realmatrix S, and to provide a constructive approach of its calculation.
In [7] the matrix S = ORO−1A−LC, where OA−LC and OR are the observability matrices of the pairs (A − LC, e1) and (R, e2) respectively. Another (more strict) condition is that the Sylvester equation SA− RS = QC, Q = SL has a unique solution S provided that the pair (A, C) is observable (in this case there exists a matrix L such that λ(A)6=λ(A−LC) =λ(R), that is equivalent to existence of a uniqueS[17]). In the present work we will apply this lemma to a nonnegative matrix R. Note that if the matrix A−LC has only real positive eigenvalues, then R can be chosen as diagonal or Jordan representation ofA−LC.
The application of Lemma 2 is connected with the Inverse eigenvalue problem for nonnegative matrices (i.e. the problem of existence of a nonnegative matrix R with the given set of eigenvaluesλ(A−LC)), see the monograph [18] (section 11.2) for the necessary and sufficient conditions which have to be imposed on λ(A−LC)in order that a nonnegative R exists. In [19] the fast Fourier transformation is used to design a real symmetricRwith a given vector of eigenvalues.
III. MAIN RESULT
Consider an LTI discrete-time system
x(t+ 1) =Ax(t) +b(t), y(t) =Cx(t) +v(t), t∈Z+, (2) where x(t)∈Rn is the state; y(t)∈Rp is the output signal available for measurements; b : Z+ → Rn, b ∈ L∞ is the input; v : Z+ → Rp, v ∈ L∞ is the measurement noise; A andC are real matrices of the corresponding dimensions.
In this section we will consider two approaches. The first one is based on a transformation of coordinates, which maps the estimation error dynamics to its cooperative representation.
The second approach is based on the error system representa- tion without transformation of coordinates.
A. Time-invariant transformation to cooperative form We will need the following assumptions.
Assumption 1. The statex(t)is bounded, i.e.x∈ L∞. Assumption 2. There exists a matrix L∈Rn×p such that
A. The matrixA−LC is Schur stable.
B. The matrixA−LC is nonnegative.
Assumption 3. Two functionsb, b:Z+→Rn,b, b∈ L∞ are given such that for all t∈Z+
b(t)≤b(t)≤b(t).
Assumption 4. The constant0≤V <+∞is given such that
||v||< V.
Assumption 1 is introduced since in this work we will not consider the problem of control design (it is rather common in the literature of the observer synthesis). The first part of Assumption 2 is standard [3], [1]. The second part is crucial for the approach, it is rather restrictive and it will be relaxed later. Assumption 3 states that the input is known up to some interval error b(t)−b(t), Assumption 4 introduces the upper bound V of the noisev amplitude.
Under the introduced assumptions an interval observer equa- tions for (2) take form:
x(t+ 1) =Ax(t) +b(t) +L(y(t)−Cx(t))−LV, (3) x(t+ 1) =Ax(t) +b(t) +L(y(t)−Cx(t)) +LV, where x(t) ∈ Rn and x(t) ∈ Rn are the lower and upper interval estimates for the state x(t),L= (L++L−)Ep×1. Theorem 1. Let assumptions 1–4 be satisfied. Then for all t∈Z+ the estimatesx(t), x(t)given by (3) are bounded and
x(t)≤x(t)≤x(t) provided that x(0)≤x(0)≤x(0).
Proof: The equation (2) can be rewritten as follows:
x(t+ 1) = (A−LC)x(t) +L[y(t)−v(t)] +b(t).
Then the dynamics of the errors e(t) =x(t)−x(t), e(t) = x(t)−x(t)obey the equations:
e(t+ 1) = (A−LC)e(t) +d(t), e(t+ 1) = (A−LC)e(t) +d(t),
where d(t) = [LV −Lv(t)] + [b(t)−b(t)], d(t) = [LV − Lv(t)] + [b(t)−b(t)]. According to assumptions 3, 4 we have d, d∈ L∞ andd(t)≥0,d(t)≥0 for allt∈Z+. Therefore, from Assumption 2.A, the variablese(t)ande(t)are bounded, and taking in mind Assumption 1 we get the boundedness of the estimatesx(t), x(t). From Assumption 2.B e(t)≥0 and e(t) ≥ 0 for all t ∈ Z+ (d, d have the same property and e(0)≥0,e(0) ≥0 by conditions), that implies the required order relation x(t)≤x(t)≤x(t)for allt∈Z+.
In order to skip the part B of Assumption 2, let us use Lemma 2.
Theorem 2. Let assumptions 1–2.A, 3–4 be satisfied, there exist matrix R ∈ Rn×n+ such that λ(A−LC) = λ(R) and the pairs(A−LC, e1),(R, e2)are observable for somee1∈ R1×n,e2∈R1×n. Then for allt∈Z+the estimatesx(t), x(t) are bounded and
x(t)≤x(t)≤x(t) provided thatx(0)≤x(0)≤x(0), where
x(t) =S+z(t)−S−z(t), x(t) =S+z(t)−S−z(t);
z(t+ 1) = Rz(t) +F y(t)−F V (4) +(S−1)+b(t)−(S−1)−b(t),
z(t+ 1) = Rz(t) +F y(t) +F V +(S−1)+b(t)−(S−1)−b(t);
z(0) = (S−1)+x(0)−(S−1)−x(0), z(0) = (S−1)+x(0)−(S−1)−x(0),
whereS=ORO−1A−LC(OA−LCandORare the observability matrices of the pairs(A−LC, e1),(R, e2)respectively),F = S−1Land F = (F++F−)Ep×1.
Proof: Consider the system (2) in the new coordinates z=S−1x:
z(t+ 1) =Rz(t) +F[y(t)−v(t)] +S−1b(t), y(t) =CSz(t) +v(t).
The dynamics of the errorse(t) =z(t)−z(t),e(t) =z(t)− z(t)obey the equations:
e(t+ 1) = Re(t) +d(t), e(t+ 1) = Re(t) +d(t),
where d(t) = [F V −F v(t)] + [S−1b(t)−(S−1)+b(t) + (S−1)−b(t)], d(t) = [F V + F v(t)] + [(S−1)+b(t) − (S−1)−b(t)−S−1b(t)]. The matrix R is Schur stable and nonnegative, thus all arguments of Theorem 1 are valid to substantiate that z(t) ≤ z(t) ≤ z(t) for all t ∈ Z+ (by construction z(0) ≤ z(0) ≤ z(0)). Next, using the relations (1) we get the theorem claim.
B. Cooperative representation in the same coordinates There is another possibility for an interval observer con- struction in the case when Assumption 2.B is not satisfied without a transformation of coordinates, but is has more complex stability conditions.
Theorem 3. Let assumptions 1, 3, 4 be satisfied and there exist a matrix L∈Rn×p such that the matrix
Σ =
D+ D−
D− D+
is Schur stable for D = A−LC. Then for all t ∈ Z+ the estimatesx(t), x(t)are bounded and
x(t)≤x(t)≤x(t) provided that x(0)≤x(0)≤x(0), where
x(t+ 1) =D+x(t)−D−x(t) +b(t) +Ly(t)−LV, (5) x(t+ 1) =D+x(t)−D−x(t) +b(t) +Ly(t) +LV, and L= (L++L−)Ep×1.
Proof: The system (2) can be rewritten as follows:
x(t+ 1) = Dx(t) +L(y(t)−v(t)) +b(t)
= [D+−D−]x(t) +L(y(t)−v(t)) +b(t).
Then the dynamics of the errors e(t) =x(t)−x(t), e(t) = x(t)−x(t)obeys the equations:
e(t+ 1) = D+e(t) +D−e(t) +d(t), (6) e(t+ 1) = D+e(t) +D−e(t) +d(t),
where d(t) = [LV −Lv(t)] + [b(t)−b(t)], d(t) = [LV − Lv(t)] + [b(t)−b(t)]. By definition, the matricesD+,D− are nonnegative. According to assumptions 3, 4 the inputs d(t), d(t)are also nonnegative, thus the estimation error dynamics is cooperative and starting frome(0)≥0,e(0)≥0we get that e(t)≥0,e(t)≥0for allt∈Z+. Since the matrixΣis Schur stable and the inputs d(t), d(t) are bounded, the variables e(t),e(t)possess the same property, that in combination with Assumption 1 implies boundedness of x(t), x(t).
It is worth doing some comments about stability verification of the matrixΣ. According to [20] a nonnegative matrixΣis Schur stable if there exists a diagonal positive definite matrix PΣ∈R2n×2n such that ΣTPΣΣ−PΣ≺0, or this condition can be replaced with existence of a vectorpΣ∈R2n+,pΣ>0 such thatpTΣ(Σ−I2n)<0. Due to block symmetric structure of the matrix Σ and its non-negativeness, the stability of Σ is equivalent to stability of the matrix D = D+ +D−. Indeed consider the copositive Lyapunov function V(t) = pTe(t) +pTe(t)for some vectorsp >0,p >0for the system (6), sincee(t)≥0,e(t)≥0for allt∈Z, then this function is positive definite. Existence of such a Lyapunov function under the condition∆V(t) =V(t+ 1)−V(t)<0,∀t∈Z, satisfied for any initial conditionse(0),e(0)∈Rn+ for the unperturbed system (6) (the system (6) withd(t) =d(t) = 0), is equivalent to its Schur stability. Thus ∆V(t) = [pTD+ +pTD− − pT]e(t) + [pTD−+pTD+−pT]e(t)and a reasonable choice is p=p, then∆V(t) = [pT{D++D−} −pT] (e(t) +e(t)), that is equivalent topT{D−In}<0 or the Schur stability of D.
Verification of stability of with simultaneous computa- tions of L and pcan be performed via the following LMIs.
By its construction, D =A−LC, then D+ = A+−L+C and D− = A−+L−C, where L+, L− are the parts of the matrixL=L+−L− with sign indefinite entries which enter into matricesD+ or D− respectively. Thus
D = D++D−=A+−L+C+A−+L−C
= A−(L+−L−)C=A−LC.
Now the nonnegative matrixDis Schur stable if there exists a diagonal positive definite matrixP such thatDTP D−P≺0.
SinceP is diagonal, then the property P0 implies that all elements of P are positive. By applying Schur complement, the matrixDstability follows the fact that there are a diagonal matrixP 0and a matrixY ∈Rn×p such that
P P A−Y C
(P A−Y C)T P
0 (7) under the constraint
P A−Y C≥0, (8)
whereL=P−1Y. Since the matrixP is elementwise positive the constraint (8) implies that the matrix D = A−LC is nonnegative, while the LMI (7) ensures the Schur stability of D. Therefore the following corollary can be formulated.
Corollary 1. Let assumptions 1, 3, 4 be satisfied and there exist a diagonal matrixP 0 and a matrixY ∈Rn×p such that (7), (8) be true. Then the result of Theorem 3 is valid.
The main advance of this corollary is that it allows us to use the numerical routines for the matrixLselection, the YALMIP toolbox of Matlab in particular can be used to find a solution of such a constrained problem [21].
Remark 1. It is worth to stress that b could be a function of the state x provided that there exist known bounded signals b, bsatisfying Assumption 3. Therefore, the presented interval observers (3) and (4) can be applied to nonlinear systems in the output canonical form, for instance. A mild reformulation of theorems 1, 2 for this case is skipped for brevity of pre- sentation. Application of these theorems to nonlinear systems is illustrated on example in Section 4.
IV. EXAMPLES
In this section we consider two examples. The first one is an academic planar system for which we will compare the interval observers proposed in theorems 2 and 3. The second example is the Hénon chaotic system.
A. Comparison of observers (4) and (5) Consider the following system
x(t+ 1) = Ax(t) +b(t), y(t) = Cx(t) +v(t),
b(t) = b0(t) +δb(t, x(t)),
0 20 40 60 80 t
−3
−2
−1 0 1 2 x2
0 20 40 60 80 t
−3
−2
−1 0 1 2
x2 b
Figure 1. Comparison of interval estimation performed by the observer (4) (part (a)) and (5) (part (b))
where x(t) ∈ R2, y(t) ∈ R, v(t) ∈ R are respectively the state, the output and the measurement noise (||v|| ≤V = 0.1, for simulation we used v(t) =V sin(t)). The signals
b0(t) = [sin(0.1t) cos(0.2t)]T, δb(t, x(t)) = δ[sin(0.5t x2(t)) sin(0.3t)]T
are the known portion ofband its uncertain deviationδbfrom the nominalb0from which we know that it is bounded byδ= 0.5. Thusb(t) =b0(t)−δ,b(t) =b0(t) +δand assumptions 3, 4 are satisfied. Finally,
A=
0.3 −0.7
0.6 −0.5
, C = [1 0],
and there is no observer gain L ∈ R2 making the matrix D =A−LC nonnegative. The matrixA is Schur stable (it has complex eigenvalues), but the matrix A = A++A− is unstable. Applying Matlab YALMIP toolbox we obtain
L= [0.3 0.6]T,
then λ(D) = [0 −0.5]T and λ(D) = [0 0.5]T, therefore all conditions of Theorem 3 are satisfied for this choice ofL. The matrices
R=
−0.5 0
1 0
, S=
0.609 0.814
−1.162 0.581
satisfy to all conditions of Theorem 2. Therefore now we can apply both observers (4) and (5) in this examples. The results of simulation for the interval estimation of the unmeasured coordinatex2 are shown in Fig. 1. As we can conclude in this examples the observers demonstrate a similar performance.
An advantage of (5) is that we can use the LMI techniques to calculate L, however performance in this case is critically dependent on ability to increase the stability margin of D by L, which is a harder problem than stabilization ofDrequired for (4). From another side, performance of the observer (4) is influenced by additional transformation of coordinates S, which also decreases the accuracy of estimation.
500 510 520 530 540 550
−3
−2
−1 0 1 2
500 510 520 530 540 550
−1
−0.5 0 0.5
t x1
x2
Figure 2. The results of simulation for Lorenz model
B. Hénon chaotic system
Consider a variant of the Hénon model:
x(t+ 1) = Ax(t) +r[1−a(t)x21(t) +d(t)], y(t) = x1(t) +v(t),
wherex(t)∈R2is the state,y(t)∈Ris the output,v(t)∈R is the measurement noise (||v|| ≤V = 0.1 and Assumption 4 is satisfied) and d(t) ∈ R is the disturbance (||d|| ≤ δ = 0.015),
A=
0 1
0.3 0
, r=
1
0
, a≤a(t)≤a, a= 1, a= 1.4.
The system has an uncertain parameter a(t) with the values from an interval. If a(t) = a, then the system equations become identical to the chaotic Hénon model. Thus assume that ||x|| < +∞ (Assumption 1 holds). Let us rewrite the system as follows:
x(t+ 1) =Ax(t) +b(t), b(t) =r[1−a(t)[y(t)−v(t)]2+d(t)]
then clearly it is in the form (2) and Assumption 3 is valid for b(t) = r[1−ay2(t)−2a|y(t)|V −aV2−δ], b(t) = r[1−ay2(t) + 2a|y(t)|V −aV2+δ], that justifies Remark 1. Finally, Assumption 2 is verified for L = [−0.1 0.1]T and C = [1 0] (the matrix A −LC is Schur stable and nonnegative). Therefore, all conditions of Theorem 1 are satisfied and the interval observer (3) solves the problem of interval state estimation. The results of simulation are shown in Fig 2.
V. CONCLUSION
The paper is devoted to interval observer design for the LTI discrete-time systems. Two techniques have been proposed.
The first one is based on a static transformation of coordi- nates, which connects a stable LTI discrete-time system with its nonnegative representation. The second technique uses a nonlinear transformation of the system in a nonnegative form, the observer gain can be calculated as a solution of LMIs. The
efficiency is shown on example of computer simulation for a chaotic system.
In comparison with continuous-time systems, the discrete- time interval observers admit a relaxation of some applicability conditions (there are more results on design of a nonnegative matrix rather than a Metzler one [18], [19]) and always there exists an interval observer (5) with a cooperative estimation error dynamics.
REFERENCES
[1] T. Meurer, K. Graichen, and E.-D. Gilles, Eds.,Control and Observer Design for Nonlinear Finite and Infinite Dimensional Systems. Springer, 2005, vol. Lecture Notes in Control and Information Sciences, no. 322.
[2] H. Nijmeijer and T. I. Fossen,New Directions in Nonlinear Observer Design. London, U.K.: Springer-Verlag, 1999.
[3] G. Besançon, Ed., Nonlinear observers and applications, ser. Lecture Notes in Control and Inforamtion Science. Berlin: Springer Verlag, 2007, vol. 363.
[4] O. Bernard and J. L. Gouzé, “Closed loop observers bundle for uncertain biotechnological models.” J. Process Control, vol. 14, pp. 765–774, 2004.
[5] M. Moisan, O. Bernard, and J. L. Gouzé, “Near optimal interval observers bundle for uncertain bio-reactors,”Automatica, vol. 45, no. 1, pp. 291–295, 2009.
[6] F. Mazenc and O. Bernard, “Asymptotically stable interval observers for planar systems with complex poles,”IEEE Trans. Autom. Control, vol. 55, no. 2, pp. 523–527, 2010.
[7] T. Raïssi, D. Efimov, and A. Zolghadri, “Interval state estimation for a class of nonlinear systems,”IEEE Trans. Automatic Control, vol. 57, no. 1, pp. 260–265, 2012.
[8] F. Mazenc and O. Bernard, “Interval observers for linear time-invariant systems with disturbances,”Automatica, vol. 47, pp. 140–147, 2011.
[9] D. Efimov, T. Raïssi, S. Chebotarev, and A. Zolghadri, “Interval state observer for nonlinear time varying systems,”Automatica, 2012.
[10] L. Jaulin, M. Kieffer, O. Didrit, and E. Walter,Applied Interval Analysis with Examples in Parameter and State Estimation, Robust Control and Robotics. Springer-Verlag, 2001.
[11] L. Chisci, A. Garulli, and G. Zappa, “Recursive state bounding by parallelotopes,”Automatica, vol. 32, no. 7, pp. 1049 – 1055, 1996.
[12] E. Walter, J. Norton, H. Piet-Lahanier, and M. Milanese, Bounding Approaches to System Identification. Perseus Publishing, 1996.
[13] D. G. Maksarov and J. P. Norton, “Computationally efficient algorithms for state estimation with ellipsoidal approximations,”Int. J. Adaptive Control and Signal Processing, vol. 16, no. 6, pp. 411–434, 2002.
[14] B. T. Polyak, S. A. Nazin, C. Durieu, and E. Walter, “Ellipsoidal parameter or state estimation under model uncertainty,” Automatica, vol. 40, no. 7, pp. 1171–1179, 2004.
[15] T. Alamo, J. M. Bravo, and E. F. Camacho, “Guaranteed state estimation by zonotopes,”Automatica, vol. 41, no. 6, pp. 1035–1043, 2005.
[16] M. W. Hirsch and H. L. Smith, “Monotone maps: a review,”J. Difference Equ. Appl., vol. 11, no. 4-5, pp. 379–398, 2005.
[17] R. H. Bartels and G. W. Stewart, “Solution of the matrix equation ax + xb = c,”Comm. ACM, vol. 15, pp. 820–826, 1972.
[18] A. Berman and R. J. Plemmons,Nonnegative matrices in the mathemat- ical sciences, ser. Classics in applied mathmatics. Philadelphia: SIAM, 1994.
[19] O. Rojo and H. Rojo, “Constructing symmetric nonnegative matrices via the fast fourier transform,”Computers and Mathematics with Applica- tions, vol. 45, pp. 1655–1672, 2003.
[20] L. Farina and S. Rinaldi,Positive Linear Systems – Theory and Appli- cations. NY: John Wiley & Sons, 2000.
[21] J. Löfberg, “Pre- and post-processing sum-of-squares programs in prac- tice,”IEEE Transactions on Automatic Control, vol. 54, no. 5, pp. 1007–
1011, 2009.
